| L(s) = 1 | − 10·7-s − 5·13-s − 8·19-s + 5·25-s + 14·31-s + 22·37-s − 13·43-s + 61·49-s + 61-s + 11·67-s − 17·73-s + 17·79-s + 50·91-s − 14·97-s + 14·103-s − 2·109-s − 22·121-s + 127-s + 131-s + 80·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 3.77·7-s − 1.38·13-s − 1.83·19-s + 25-s + 2.51·31-s + 3.61·37-s − 1.98·43-s + 61/7·49-s + 0.128·61-s + 1.34·67-s − 1.98·73-s + 1.91·79-s + 5.24·91-s − 1.42·97-s + 1.37·103-s − 0.191·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 6.93·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8283763564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8283763564\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.035788711001912166293579334570, −8.869174687565740640318528976980, −8.167951033670281401432011554682, −8.003470138624616691126597580646, −7.34446661217693989895547168690, −6.87866226090585347534914510805, −6.57889647606944001186049704635, −6.53153397749464457925081429892, −6.05570793779726981709423426598, −5.87694370711244336982328164179, −5.07504469918525753299014565658, −4.64212720473546889163396201299, −4.09529705386196978513541226870, −3.94327790364491916845389763369, −3.03544499458227196949544199042, −2.96060964600432638123656321031, −2.64915669376662603274597772041, −2.15074744096389971466183779754, −0.791239675402527125698551438652, −0.40760867180836515771508308078,
0.40760867180836515771508308078, 0.791239675402527125698551438652, 2.15074744096389971466183779754, 2.64915669376662603274597772041, 2.96060964600432638123656321031, 3.03544499458227196949544199042, 3.94327790364491916845389763369, 4.09529705386196978513541226870, 4.64212720473546889163396201299, 5.07504469918525753299014565658, 5.87694370711244336982328164179, 6.05570793779726981709423426598, 6.53153397749464457925081429892, 6.57889647606944001186049704635, 6.87866226090585347534914510805, 7.34446661217693989895547168690, 8.003470138624616691126597580646, 8.167951033670281401432011554682, 8.869174687565740640318528976980, 9.035788711001912166293579334570
